Fig.  REF.
Explanation of the optimal permutation problem.
a Correlation matrix of the neuronal activity of the C. elegans.
Each entry \(C_{ij}\)  is the correlation coefficient between the time
series \(x_i^t\)  and \(x_j^t\)  measuring the temporal activities of
neurons \(i\)  and \(j\)  (data are from Ref. [1]}).
b Two blocks filter \(A\)  to be applied to matrix \(B\)  to perform
the clustering of \(B_{ij}\)  into 2 blocks, each one made up of
neurons maximally correlated among each other. The two blocks
are arranged along the main diagonal.
c Three blocks filter which, similarly to the filter in b, produces
a clustering of \(B\)  into 3 clusters.
d Minimization of the cost function \(E(P)\)  defined in
Equation (REF ) for two and three-blocks filters.
e The optimal permutation \(P_*\)  that solves the OPP defined
in Equation (REF ) for the correlation matrix
\(B\)  shown in a and the two-blocks filter \(A\)  in b.
The permutation \(P_*\)  is the one that minimizes the cost function
in d (red curve), i.e., \(P_*: \min _{P}E(P) = E(P_*)\) .
f Clustered correlation matrix \(B^{\prime }=P_*^{\rm t}BP_*\)  obtained
by solving the OPP with a two-blocks filter (left panel) and a
three-blocks filter (right panel).
